Mathematics

This course covers current topics in Combinatorics. More specific topic details are provided when the course is offered.

Introduction to research in computational complexity theory. Computational models: nondeterministic, alternating, and probabilistic machines. Boolean circuits. Complexity classes associated with these models: NP, Polynomial hierarchy, BPP, P/poly, etc. Complete problems. Interactive proof systems and probabilistically checkable proofs: IP=PSPACE and NP=PCP (log n, 1). Definitions of randomness. Pseudorandomness and derandomizations. Lower bounds for concrete models such as algebraic decision trees, bounded-depth circuits, and monotone circuits.

This course covers current topics in Discrete Mathematics. Specific topic information provided when the course is taught.

This course focuses on spectral methods useful in the analysis of big data sets. Spectral methods involve the construction of matrices (or linear operators) directly from the data and the computation of a few leading eigenvectors and eigenvalues for information extraction. Examples include the singular value decomposition and the closely related principal component analysis; the PageRank algorithm of Google for ranking web sites; and spectral clustering methods that use eigenvectors of the graph Laplacian.

This course focuses on computational methods in cryo-EM, including three-dimensional ab-initio modelling, structure refinement, resolving structural variability of heterogeneous populations, particle picking, model validation, and resolution determination. Special emphasis is given to methods that play a significant role in many other data science applications. These comprise of key elements of statistical inference, image processing, and linear and non-linear dimensionality reduction. The software packages RELION and ASPIRE are routinely used for class demonstration on both simulated and publicly available experimental datasets.

This course covers current topics in Ergodic Theory. More specific topic details provided when course is offered.

This course covers current topics in Probability, Statistics and Dynamics. More specific topic details provided when the course is offered.

The course covers current topics in Mathematical Physics. More specific topic details provided when the course is offered.

An introduction to mathematically rigorous methods in physics. Topics to be covered include classical and quantum statistical mechanic, quantum many-body problem, group theory, Schroedinger operators, and quantum information theory.